Completeness in quasi-uniform spaces

Andrikopoulos, Athanasios

doi:10.1023/b:amhu.0000045540.39991.f2

Cited by 8 publications

(9 citation statements)

References 18 publications

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“…On the other hand, since the space (X, U) is half-complete, there exists ξ ∈ X such that (φ(x γ )) γ∈Γ τ (U )-converges to ξ (2). Hence by (1) and (2) we conclude that {φ(y(a k , W k ))|(a k , W k ) ∈ A ⋆ , k ∈ K} τ (U )-converges to ξ. Since {φ(y(a k , W k ))|(a k , W k ) ∈ A ⋆ , k ∈ K} is a subnet of φ(y(a, W )) (a,W )∈A ⋆ we conclude that ξ is a cluster point of the latter.…”

Section: Definition 31 (See [6 Page 81])mentioning

confidence: 79%

∗-half completeness in quasi-uniform spaces

Andrikopoulos

2009

Appl.Gen.Topol.

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Abstract.Romaguera and have introduced the notion of T1 * -half completion and used it to see when a quasi-uniform space has a * -compactification. In this paper, for any quasi-uniform space, we construct a * -half completion, called standard * -half completion. The constructed * -half completion coincides with the usual uniform completion in the uniform spaces and is the unique (up to quasi-isomorphism) T1* -half completion of a symmetrizable quasi-uniform space. Moreover, it constitutes a * -compactification for * -Cauchy bounded quasi-uniform spaces. Finally, we give an example which shows that the standard * -half completion differs from the bicompletion construction.2000 AMS Classification: 54E15, 54D35.

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Section: Definition 31 (See [6 Page 81])mentioning

confidence: 79%

∗-half completeness in quasi-uniform spaces

Andrikopoulos

2009

Appl.Gen.Topol.

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“…The quasi-uniform space (X, U ) is called D-complete provided every D-Cauchy filter converges. A related notion of completeness was considered by Andrikopoulos [12]. For a comparative study of the completeness notions defined by filters and nets see Andrikopoulos [13], Deák [38,39,40] and Sünderhauf [168,169].…”

Section: 3mentioning

confidence: 99%

Functional Analysis in Asymmetric Normed Spaces

Cobzaş

2013

Frontiers in Mathematics

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171

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The aim of this paper is to present a survey of some recent results obtained in the study of spaces with asymmetric norm. The presentation follows the ideas from the theory of normed spaces (topology, continuous linear operators, continuous linear functionals, duality, geometry of asymmetric normed spaces, compact operators) emphasizing similarities as well as differences with respect to the classical theory. The main difference comes form the fact that the dual of an asymmetric normed space X is not a linear space, but merely a convex cone in the space of all linear functionals on X. Due to this fact, a careful treatment of the duality problems (e.g. reflexivity) and of other results as, for instance, the extension of fundamental principles of functional analysis -the open mapping theorem and the closed graph theorem -to this setting, is needed. Contents 1. Introduction: 1.1 Quasi-metric spaces and asymmetric normed spaces; 1.2 The topology of a quasisemimetric space; 1.3 Quasi-uniform spaces.2. Completeness and compactness in quasi-metric and in quasi-uniform spaces: 2.1 Various notions of completeness for quasi-metric spaces; 2.2 Compactness, total bondedness and precompactness; 2.3 Completeness in quasi-uniform spaces; 2.4 Baire category.3. Continuous linear operators between asymmetric normed spaces: 3.1 The asymmetric norm of a continuous linear operator; 3.2 The normed cone of continuous linear operators -completeness; 3.3 The bicompletion of an asymmetric normed space; 3.4 Open mapping and closed graph theorems for asymmetric normed spaces; 3.5 Normed cones; 3.6 The w ♭ topology of the dual space X ♭ p ; 3.7 Compact subsets of an asymmetric normed space; 3.8 The conjugate operator, precompact operators between asymmetric normed spaces and a Schauder type theorem; 3.9 Asymmetric moduli of smoothness and rotundity; 3.10 Asymmetric topologies on normed lattices. 4. Linear functionals on an asymmetric normed space: 4.1 Some properties of continuous linear functionals; 4.2 Hahn-Banach type theorems; 4.3 The bidual space, reflexivity and Goldstine theorem. 5. The Minkowski functional and the separation of convex sets: 5.1 The Minkowski gauge functional -definition and properties; 5.2 The separation of convex sets; 5.3 Extremal points and Krein-Milman theorem.6. Applications to best approximation: 6.1 Characterizations of nearest points in convex sets and duality; 6.2 The distance to a hyperplane; 6.3 Best approximation by elements of sets with convex complement; 6.4 Optimal points; 6.5 Sign-sensitive approximation in spaces of continuous or integrable functions.7. Spaces of semi-Lipschitz functions: 7.1 Semi-Lipschitz functions -definition, the extension property, applications to best approximation in quasi-metric spaces; 7.2 Properties of the cone of semi-Lipschitz functions -linearity, completeness. ReferencesThis research was supported by Grant CNCSIS 2261, ID 543.Remark 1.1. Since the terms "quasi-norm", "quasi-normed space" and "quasi-Banach space" are already "registered trademarks" (see, for instance, th...

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“…Lemma 2.25 (see [13,Proposition 4.8]). Let (X,ᐁ) be a quasi-uniform space, let Ᏺ be a minimal ᐁ -Cauchy filter on it and let (g (x,F) ) (x,F)∈Λ , (g (y,F) ) (y,F)∈K be two associated nets of Ᏺ.…”

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confidence: 99%

“…Lemma 2.26 (see [13,Proposition 4.7]). If two minimal ᐁ -Cauchy filters in a quasiuniform space (X,ᐁ) have a common associated net, then they coincide.…”

Section: )mentioning

confidence: 99%

Symmetry Conditions on the Coincidence of Some Notions of Quasi-Uniform Completeness

Andrikopoulos¹

2007

International Journal of Mathematics and Mathematical Sciences

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We generalize the notions of quietness and semisymmetry defined by Doitchinov (1991) and Deák (1991) and we study the role of these extended notions on the coincidence of some well-known quasi-uniform completeness. In particular, it is shown that the bicompletion coincides (up to quasi-uniformism) with the standard D-completion in quiet -weakly pair Cauchy bounded quasi-uniform spaces and it coincides with -half-completion defined by Romaguera and Sánchez-Granero (2002), in T 0 -weakly quiet quasiuniform spaces. definition of Cauchy net or Cauchy filter (often nets and filters lead to equivalent theories). The reason for the difficulty to develop a satisfactory theory of completeness for the class of all quasi-uniform spaces is due to the difficulty to approach the notion of Cauchy net (filter) in an appropriate way from the case of uniform spaces to the case of quasiuniform spaces. More precisely, since uniform spaces belong to the class of quasi-uniform spaces, according to Doitchinov [1], a notion of Cauchy net in any quasi-uniform space has to be defined in such a manner that this definition provides the properties that convergent nets are Cauchy, and it agrees with the usual definition for uniform spaces. Moreover, the suggested completion must be a monotone operator with respect to inclusion and give rise to the usual uniform completion in the uniform case.Fletcher and Lindgren in [4,5] give two equivalent theories of completeness, the pair completeness or bicompleteness. The first one uses pair of filters (Ᏺ,Ᏻ) for which Ᏺ ∪ Ᏻ is a filter base and the second is modelled upon the presentations of the completion of a uniform space given by Bourbaki [6]. It turns out that this concept coincides (for quasiuniform spaces) with that of double completeness introduced by Császár in [7]. The authors proved that the bicompletion has a very nice behavior: every T 0 quasi-uniform space can be quasi-uniformly embedded in a unique (i.e., up to quasi-uniformism) T 0 quasi-uniform space; for uniform spaces the construction coincides with the usual one. In [1], Doitchinov has introduced the notion of D-completeness and he proved that a certain category of quasi-uniform T 2 -spaces, the so-called quiet spaces, has a nice behavior: every quiet T 2 -space can be quasi-uniformly embedded in a unique D-complete quiet T 2 -space; for uniform spaces (which are always quiet) the construction coincides with the usual one. D-completeness is founded on a concept of a Cauchy net which can be done satisfying some most natural basic requirements. The Doitchinov completion provides another sophisticated completion that is in general larger than the bicompletion. (Consider for instance the set of the rationals equipped with the Sorgenfrey quasimetric. The space is bicomplete and only its Doitchinov completion yields the expected completion.)The definition of Cauchy net or filter in quasi-uniform space which has been proposed in bicompleteness and D-completeness gives a completing theory which satisfies the above requirements given by Doit...

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Completeness in quasi-uniform spaces

Cited by 8 publications

References 18 publications

∗-half completeness in quasi-uniform spaces

∗-half completeness in quasi-uniform spaces

Functional Analysis in Asymmetric Normed Spaces

Symmetry Conditions on the Coincidence of Some Notions of Quasi-Uniform Completeness

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